Is there a way to display the view of an active program/screen as a texture on a 3D object?
Rotating live data out of view is what it sounds like you are after. 

You can create a screenshot of the static elements you want to animate and turn them into a gif from frame by frame files. But live data would not be possible, not without extending blender with a new plugin.  Even then, the best place for this kind of transformation to take place would be within rainmeter.  

You may want to approach the rainmeter community with questions about how to take a screen cap, and use lua to apply rotation transforms on a flat screen capped image.  

Hope that helps.